Oppenheim's algebraicity conjecture for linear forms

Let $n\geq 3$, and let $\langle\mathbf{L}_1,\,\cdot\,\rangle,\dots,\langle\mathbf{L}_n,\,\cdot\,\rangle$ be $n$ linearly independent linear forms on $\mathbb{R}^n$ satisfying

$$\inf_{\mathbf{x}\in\mathbb{Z}^n\setminus\{\boldsymbol{0}\}}\left|\langle\mathbf{L}_1,\mathbf{x}\rangle\dots\langle\mathbf{L}_n,\mathbf{x}\rangle\right|>0.$$

Oppenheim's conjecture. The lattice

$$\left\{(\langle\mathbf{L}_1,\mathbf{x}\rangle,\dots,\langle\mathbf{L}_n,\mathbf{x}\rangle)\mid\mathbf{x}\in\mathbb{Z}^n\right\}$$

is algebraic, meaning that it is similar, modulo the action of the group of diagonal $n\times n$ matrices, to the lattice of a complete module of a totally real algebraic number field of degree $n$. The source presents this as a classical conjecture and states that it remains unproved.